10. Appendix

646 Appendix B
electron is close to the core where V is large the electron wave function ˇ
will be varying rapidly over a distance which is of the order of the Bohr radius.
On the other hand, (1/k) is of the order of the lattice constant which is
much larger than the Bohr radius. Therefore 〈ˇ|p|ˇ〉 ∼〈ˇ|∇|ˇ〉 tends to be
much larger than k (see, for example, E.O. Kane: Energy band structure in
p-type germanium and silicon. J. Phys. Chem. Solids 1, 83 (1956)). When the
second term is neglected we obtain again (2.45a) for the S-O coupling. However,
for a general point k in the Brillouin Zone one finds the k · p term to be
the dominant interaction which mixes the k 0 electron states so the effect
of HSO on the electrons is no longer discernible. There are two exceptional
cases though. They are when k is along the [111] or [100] directions.
In Sections 2.4.1 and 2.4.2 we learn that the group of § in both the
zincblende and diamond structure has three-fold rotational symmetry about
the [111] axis. This symmetry is high enough that in the diamond-type semiconductor
the triply degenerate p-like wave functions at k 0 splits into a
doublet with symmetry §3 and a singlet with symmetry §1. See Fig. 2.10 for
the band structure in Si neglecting S-O splitting and Table 2.12 for the character
table (same for diamond and zincblende crystals along the [111] direction
as long as k is inside the Brillouin zone). This degeneracy of the §3 state can
be split by the S-O coupling.
In fact, we can model the valence band electron in this case as a “p-like”
electron in a cylindrical potential. In case of the L 1 state we expect the
three degenerate states to be split into a doublet with Lz ±1 and a singlet
with Lz 0 We will assume that the axis of quantization of the angular
momentum vector L to be the [111] axis which will be labeled as the z-axis
to simplify the notation. We can identify the doubly degenerate §3 states as
corresponding to the Lz ±1 states. The wave functions for the §3 state can
then be represented as [|X〉 i|Y〉]/ √ 2 and [|X〉 i|Y〉]/ √ 2 to correspond to
the Lz ±1 states. Again |X〉 and |Y〉 are shorthand notations for the two §3
wave functions which transform into each under the symmetry operations of
the group of § like the spatial coordinates x and y. In this coordinate system
the S-O interaction is given by: HSO Ï ′ L · S Ï ′ LzSz. We will assume at
first that the spin-orbit coupling constant Ï ′ is not the same as Ï. We can show
that:
〈Lz 1, Sz 1/2|HSO|Lz 1, Sz 1/2〉 Ï ′ /2 while
〈Lz 1, Sz 1/2|HSO|Lz 1, Sz 1/2〉 Ï ′ /2.
Similarly 〈Lz 1, Sz 1/2|HSO|Lz 1, Sz 1/2〉 Ï ′ /2 while
〈Lz 1, Sz 1/2|HSO|Lz 1, Sz 1/2〉 Ï ′ /2.
In other words, along the [111] direction the Jz ±3/2 states are split
from the Jz ±1/2 states by the spin-orbit coupling. The magnitude of the
S-O coupling ¢1 is equal to Ï ′ .IfÏ Ï ′ then we obtain the “two-thirds rule”:
¢1/¢0 2/3.
The reason why one may expect Ï to differ from Ï ′ is that the k · p perturbation
term mixes the k 0 wave functions. For example, the k · p term
will mix the anti-bonding °15 conduction band wave function with the bonding